Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2104.01941

A Note on Enumeration by Fair Sampling

Yuta Mizuno, Tamiki Komatsuzaki

Published Apr 5, 2021• 0 citations
arXivHTMLPDF

Abstract

This note describes an algorithm for enumerating all the elements in a finite set based on uniformly random sampling from the set. This algorithm can be used for enumeration by fair sampling with quantum annealing. Our algorithm is based on a lemma of the coupon collector's problem and is an improved version of the algorithm described in arXiv:2007.08487 (2020). We provide a mathematical analysis and a numerical demonstration of our algorithm.

DOI
10.48550/arxiv.2104.01941

Related papers

Quantum Coupon Collector

Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs +3 more

We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω(k), then Ω(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

arXiv:2002.07688•Leibniz international proceedings in informatics•Jan 1, 2020•9 citations

Fair Sampling by Simulated Annealing on Quantum Annealer

Masayuki Yamamoto, Masayuki Ohzeki, Kazuyuki Tanaka

Conventional quantum annealing does not sample all ground states fairly. We demonstrate that fair sampling can be achieved by performing simulated annealing on a quantum annealer. We discuss the problems that occur when implementing this method and propose an alternative way to overcome them. We numerically verify the fair sampling ability of our method in a small-scale toy model.

arXiv:1912.10701•Journal of the Physical Society of Japan•Jan 28, 2020•34 citations

Achieving fair sampling in quantum annealing

Vaibhaw Kumar, Casey Tomlin, C. Nehrkorn +2 more

Sampling all ground states of a Hamiltonian with equal probability is a desired feature of a sampling algorithm, but recent studies indicate that common variants of transverse field quantum annealing sample the ground state subspace unfairly. In this note, we present perturbation theory arguments suggesting that this deficiency can be corrected by employing reverse annealing-inspired paths. We confirm that this conclusion holds in simulations of previously studied small instances with degeneracy, as well as larger instances on quantum annealing hardware.

arXiv:2007.08487•Jul 16, 2020•12 citations

Enumeration algorithms for combinatorial problems using Ising machines: When should we stop exploring energy landscapes?

Yuta Mizuno, Ali, Mohammad, Tamiki Komatsuzaki

Combinatorial problems such as combinatorial optimization and constraint satisfaction problems arise in decision-making across various fields of science and technology. In real-world applications, when multiple optimal or constraint-satisfying solutions exist, enumerating all these solutions is often desirable, as it provides flexibility in decision-making. However, combinatorial problems and their enumeration versions pose significant computational challenges due to combinatorial explosion. To address these challenges, we propose enumeration algorithms for combinatorial optimization and constraint satisfaction problems using Ising machines. Ising machines are specialized devices designed to efficiently solve combinatorial problems by exploring the energy landscape of an Ising model. Ising machines typically sample lower-energy solutions with higher probability. Our enumeration algorithms repeatedly perform such sampling to collect all desirable solutions. The crux of the proposed algorithms lies in their stopping criteria for sampling-based energy landscape exploration, which are derived from probability theory. In particular, the proposed algorithms have theoretical guarantees that the failure probability of enumeration is bounded above by a user-specified value, provided that lower-cost solutions are sampled more frequently and equal-cost solutions are sampled with equal probability. Many physics-based Ising machines are expected to (approximately) satisfy these conditions. As a demonstration, we applied our algorithm using simulated annealing to maximum clique enumeration on random graphs. We found that our algorithm enumerates all maximum cliques in large, dense graphs faster than a conventional branch-and-bound algorithm specifically designed for maximum clique enumeration. These findings underscore the effectiveness and potential of our proposed approach.

arXiv:2412.00284•Nov 29, 2024•1 citation

Fair sampling of ground-state configurations using hybrid quantum-classical MCMC algorithms

Yuichiro Nakano, Keisuke Fujii

We study the fair sampling properties of hybrid quantum-classical Markov chain Monte Carlo (MCMC) algorithms for combinatorial optimization problems with degenerate ground states. While quantum optimization heuristics such as quantum annealing and the quantum approximate optimization algorithm (QAOA) are known to induce biased sampling, hybrid quantum-classical MCMC incorporates quantum dynamics only as a proposal transition and enforces detailed balance through classical acceptance steps. Using small Ising models, we show that MCMC post-processing corrects the sampling bias of quantum dynamics and restores near-uniform sampling over degenerate ground states. We then apply the method to random $k$-SAT problems near the satisfiability threshold. For random 2-SAT, a hybrid MCMC combining QAOA-assisted neural proposals with single spin-flip updates achieves fairness comparable to that of PT-ICM. For random 3-SAT, where such classical methods are no longer applicable, the hybrid MCMC still attains approximately uniform sampling. We also examine solution counting and find that the required number of transitions is comparable to that of WalkSAT. These results indicate that hybrid quantum-classical MCMC provides a viable framework for fair sampling and solution enumeration.

quant-pharXiv:2512.14552•Dec 17, 2025•0 citations

Uncertain fate of fair sampling in quantum annealing

Mario S. Könz, Guglielmo Mazzola, Andrew J. Ochoa +2 more

Recently, it was demonstrated both theoretically and experimentally on the D-Wave quantum annealer that transverse-field quantum annealing does not find all ground states with equal probability. In particular, it was proposed that more complex driver Hamiltonians beyond transverse fields might mitigate this shortcoming. Here, we investigate the mechanisms of (un)fair sampling in quantum annealing. While higher-order terms can improve the sampling for selected small problems, we present multiple counterexamples where driver Hamiltonians that go beyond transverse fields do not remove the sampling bias. Using perturbation theory we explain why this is the case. In addition, we present large-scale quantum Monte Carlo simulations for spin glasses with known degeneracy in two space dimensions and demonstrate that the fair-sampling performance of quadratic driver terms is comparable to standard transverse-field drivers. Our results suggest that quantum annealing machines are not well suited for sampling applications, unless postprocessing techniques to improve the sampling are applied.

arXiv:1806.06081•Physical Review A•Sep 20, 2019•47 citations
This note describes an algorithm for enumerating all the elements in a finite set based on uniformly random sampling from the set. This algorithm can be used for enumeration by fair sampling with quantum annealing. Our algorithm is based on a lemma of the coupon collector's problem and is an improved version of the algorithm described in arXiv:2007.08487 (2020). We provide a mathematical analysis and a numerical demonstration of our algorithm.
We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω(k), then Ω(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.
Sampling all ground states of a Hamiltonian with equal probability is a desired feature of a sampling algorithm, but recent studies indicate that common variants of transverse field quantum annealing sample the ground state subspace unfairly. In this note, we present perturbation theory arguments suggesting that this deficiency can be corrected by employing reverse annealing-inspired paths. We confirm that this conclusion holds in simulations of previously studied small instances with degeneracy, as well as larger instances on quantum annealing hardware.
Combinatorial problems such as combinatorial optimization and constraint satisfaction problems arise in decision-making across various fields of science and technology. In real-world applications, when multiple optimal or constraint-satisfying solutions exist, enumerating all these solutions is often desirable, as it provides flexibility in decision-making. However, combinatorial problems and their enumeration versions pose significant computational challenges due to combinatorial explosion. To address these challenges, we propose enumeration algorithms for combinatorial optimization and constraint satisfaction problems using Ising machines. Ising machines are specialized devices designed to efficiently solve combinatorial problems by exploring the energy landscape of an Ising model. Ising machines typically sample lower-energy solutions with higher probability. Our enumeration algorithms repeatedly perform such sampling to collect all desirable solutions. The crux of the proposed algorithms lies in their stopping criteria for sampling-based energy landscape exploration, which are derived from probability theory. In particular, the proposed algorithms have theoretical guarantees that the failure probability of enumeration is bounded above by a user-specified value, provided that lower-cost solutions are sampled more frequently and equal-cost solutions are sampled with equal probability. Many physics-based Ising machines are expected to (approximately) satisfy these conditions. As a demonstration, we applied our algorithm using simulated annealing to maximum clique enumeration on random graphs. We found that our algorithm enumerates all maximum cliques in large, dense graphs faster than a conventional branch-and-bound algorithm specifically designed for maximum clique enumeration. These findings underscore the effectiveness and potential of our proposed approach.
We study the fair sampling properties of hybrid quantum-classical Markov chain Monte Carlo (MCMC) algorithms for combinatorial optimization problems with degenerate ground states. While quantum optimization heuristics such as quantum annealing and the quantum approximate optimization algorithm (QAOA) are known to induce biased sampling, hybrid quantum-classical MCMC incorporates quantum dynamics only as a proposal transition and enforces detailed balance through classical acceptance steps. Using small Ising models, we show that MCMC post-processing corrects the sampling bias of quantum dynamics and restores near-uniform sampling over degenerate ground states. We then apply the method to random kkk-SAT problems near the satisfiability threshold. For random 2-SAT, a hybrid MCMC combining QAOA-assisted neural proposals with single spin-flip updates achieves fairness comparable to that of PT-ICM. For random 3-SAT, where such classical methods are no longer applicable, the hybrid MCMC still attains approximately uniform sampling. We also examine solution counting and find that the required number of transitions is comparable to that of WalkSAT. These results indicate that hybrid quantum-classical MCMC provides a viable framework for fair sampling and solution enumeration.
Recently, it was demonstrated both theoretically and experimentally on the D-Wave quantum annealer that transverse-field quantum annealing does not find all ground states with equal probability. In particular, it was proposed that more complex driver Hamiltonians beyond transverse fields might mitigate this shortcoming. Here, we investigate the mechanisms of (un)fair sampling in quantum annealing. While higher-order terms can improve the sampling for selected small problems, we present multiple counterexamples where driver Hamiltonians that go beyond transverse fields do not remove the sampling bias. Using perturbation theory we explain why this is the case. In addition, we present large-scale quantum Monte Carlo simulations for spin glasses with known degeneracy in two space dimensions and demonstrate that the fair-sampling performance of quadratic driver terms is comparable to standard transverse-field drivers. Our results suggest that quantum annealing machines are not well suited for sampling applications, unless postprocessing techniques to improve the sampling are applied.